| Black hole evaporation |
|
||
| A brief analysis of the mathematical results of Hawking radiation. | |||
|
|
|||
| Black hole evaporation. | |||
| The Hawking temperature T of a Schwarzschild (nonrotating, uncharged) black hole with mass m is given by the equation (in geometrized units) [reference 1] | |||
| T = hbar/(8 pi k m). | equation 1 | ||
| In conventional units (which we use here), this would be written | |||
| T = (hbar c3)/(8 pi G k m). | equation 2 | ||
|
The emission of this energy results in an energy decrease of the black
hole, and thus a loss in its mass. What period of time tau will it
take for a black hole of mass mu to evaporate completely? A black hole with mass m has a Schwarzschild radius |
|||
| r = 2 G m/c2 | equation 3 | ||
| and thus an area of | |||
| A = 4 pi r2 | equation 4 | ||
| A = 16 pi G2 m2/c4. | equation 5 | ||
| Hawking radiation would have a power P related to the hole's area A and its temperature T by the blackbody power law (with e = 1), | |||
| P = sigma A T4 | equation 6 | ||
| P = (sigma hbar4 c8)/(256 pi3 G2 k4 m2) | equation 7 | ||
| or more conveniently, | |||
| P = K/m2 | equation 8 | ||
| where K == (sigma hbar4 c8)/(256 pi3 G2 k4) = 3.563 x 1032 W kg2. Given that the power of the Hawking radiation is the rate of energy loss of the hole, we can write | |||
| P = -dE/dt. | equation 9 | ||
| Since the total energy E of the hole is related to its mass m by Einstein's mass-energy formula, | |||
| E = m c2 | equation 10 | ||
| we can then rewrite P = -dE/dt as | |||
| P = -(d/dt) (m c2) | equation 11 | ||
| P = -c2 dm/dt. | equation 12 | ||
| We can then equate this to our above expression for the power, P = K/m2, and find | |||
| -c2 dm/dt = K/m2. | equation 13 | ||
| This differential equation is separable, and we can write | |||
| m2 dm = -K/c2 dt. | equation 14 | ||
| Integrating over m from mu (the initial mass of the hole) to zero (complete evaporation), and over t from zero to tau, we find that | |||
| tau = c2/(3 K) mu3. | equation 15 | ||
|
That is, the evaporation time of the hole is proportional to the cube
of its mass.
|
|||
| References. | |||
|
1. Black holes, white dwarfs, and neutron stars: The physics of compact objects Stuart L. Shapiro, Saul A. Teukolsky p. 366 Wiley-Interscience; 1983 |
reference 1 | ||
| Navigation. | |||
|
Erik Max Francis
-- TOP Welcome to my homepage. |
|
||
|
Writing
-- UP Various things I've written. |
|
||
|
Essays
-- UP Essays I've written. |
|
||
|
Geosynchronous and geostationary orbits
-- PREVIOUS |
|
||
|
Gravitational redshift
-- NEXT A thought-experiment demonstrating the existence of gravitational redshift. |
|
||
| Quick links. | |||
|
Contents of Erik Max Francis' homepages
-- CONTENTS Everything in my homepages. |
|
||
|
Feedback
-- FEEDBACK How to send feedback on these pages to the author. |
|
||
|
About Erik Max Francis
-- PERSONAL Information about me. |
|
||
|
Copyright
-- COPYRIGHT Copyright information regarding these pages. |
|
||
|
|
|||
| Copyright © 1995 Erik Max Francis. All rights reserved. |
|
||
|